what's a piece of land used as an extra garden called?

I think the word you're looking for is "allotment" or allotment garden.

In the US I think people would describe it as a form of 'community garden'.

It's basically a piece of land that has been "allotted" (given or appropriated) to someone, in order for them to grow plants or vegetables on.

I don't think it's a very common concept, which could be why you were struggling to find the word for it..

I only know what one is because of the British TV show Eastenders, who conveniently use the concept of an allotment every time a garden background is required for a scene.

I mentioned "community garden", but it is worth pointing out that most uses of the term community garden refer to an area that is shared or tended to by a community - whereas allotments are often partitioned, with specific parts of the garden allocated to individuals.